Convergence analysis of Markov chain Monte Carlo methods in high-dimensional statistical applications is increasingly recognized. In this paper, we develop general mixing time bounds for Metropolis-Hastings algorithms on discrete spaces by building upon and refining some recent theoretical advancements in Bayesian model selection problems. We establish sufficient conditions for a class of informed Metropolis-Hastings algorithms to attain relaxation times that are independent of the problem dimension. These conditions are grounded in high-dimensional statistical theory and allow for possibly multimodal posterior distributions. We obtain our results through two independent techniques: the multicommodity flow method and single-element drift condition analysis; we find that the latter yields a tighter mixing time bound. Our results and proof techniques are readily applicable to a broad spectrum of statistical problems with discrete parameter spaces.

Generating graphs that preserve characteristic structures while promoting sample diversity can be challenging, especially when the number of graph observations is small. Here, we tackle the problem of graph generation from only one observed graph. The classical approach of graph generation from parametric models relies on the estimation of parameters, which can be inconsistent or expensive to compute due to intractable normalisation constants. Generative modelling based on machine learning techniques to generate high-quality graph samples avoids parameter estimation but usually requires abundant training samples. Our proposed generating procedure, SteinGen, which is phrased in the setting of graphs as realisations of exponential random graph models, combines ideas from Stein's method and MCMC by employing Markovian dynamics which are based on a Stein operator for the target model. SteinGen uses the Glauber dynamics associated with an estimated Stein operator to generate a sample, and re-estimates the Stein operator from the sample after every sampling step. We show that on a class of exponential random graph models this novel "estimation and re-estimation" generation strategy yields high distributional similarity (high fidelity) to the original data, combined with high sample diversity.

Federated learning (FL) is an approach to training machine learning models that takes advantage of multiple distributed datasets while maintaining data privacy and reducing communication costs associated with sharing local datasets. Aggregation strategies have been developed to pool or fuse the weights and biases of distributed deterministic models; however, modern deterministic deep learning (DL) models are often poorly calibrated and lack the ability to communicate a measure of epistemic uncertainty in prediction, which is desirable for remote sensing platforms and safety-critical applications. Conversely, Bayesian DL models are often well calibrated and capable of quantifying and communicating a measure of epistemic uncertainty along with a competitive prediction accuracy. Unfortunately, because the weights and biases in Bayesian DL models are defined by a probability distribution, simple application of the aggregation methods associated with FL schemes for deterministic models is either impossible or results in sub-optimal performance. In this work, we use independent and identically distributed (IID) and non-IID partitions of the CIFAR-10 dataset and a fully variational ResNet-20 architecture to analyze six different aggregation strategies for Bayesian DL models. Additionally, we analyze the traditional federated averaging approach applied to an approximate Bayesian Monte Carlo dropout model as a lightweight alternative to more complex variational inference methods in FL. We show that aggregation strategy is a key hyperparameter in the design of a Bayesian FL system with downstream effects on accuracy, calibration, uncertainty quantification, training stability, and client compute requirements.

Gaussian process (GP) regression is a non-parametric, Bayesian framework to approximate complex models. Standard GP regression can lead to an unbounded model in which some points can take infeasible values. We introduce a new GP method that enforces the physical constraints in a probabilistic manner. This GP model is trained by the quantum-inspired Hamiltonian Monte Carlo (QHMC). QHMC is an efficient way to sample from a broad class of distributions. Unlike the standard Hamiltonian Monte Carlo algorithm in which a particle has a fixed mass, QHMC allows a particle to have a random mass matrix with a probability distribution. Introducing the QHMC method to the inequality and monotonicity constrained GP regression in the probabilistic sense, our approach improves the accuracy and reduces the variance in the resulting GP model. According to our experiments on several datasets, the proposed approach serves as an efficient method as it accelerates the sampling process while maintaining the accuracy, and it is applicable to high dimensional problems.

The notion of inference on function spaces is essential to the physical sciences and engineering, where the governing equations are frequently partial differential equations (PDEs) describing the evolution of functions in space and time. In particular, it is often desirable to infer the values of a function everywhere in a physical domain given a sparse number of observation points. There are numerous types of problems in which functional regression plays an important role, such as inverse problems, time series forecasting, data imputation/assimilation. Functional regression problems can be particularly challenging for real world datasets because the underlying stochastic process is often unknown. Much of the work on functional regression and inference has relied on Gaussian processes (GPs) (Rasmussen and Williams, 2006), a specific type of stochastic process in which any finite collection of points has a multivariate Gaussian distribution. Some of the earliest applications focused on analyzing geological data, such as the locations of valuable ore deposits, to identify where new deposits might be found (Chiles and Delfiner, 2012). GP regression (GPR) provides several advantages for functional inference including robustness and mathematical tractability for various problems. This has led to the use of GPR in an assortment of scientific and engineering fields, where precision and reliability in predictions and inferences can significantly impact outcomes (Deringer et al., 2021; Aigrain and Foreman-Mackey, 2023). Despite widespread adoption, the assumption of a GP prior for functional inference problems can be rather limiting, particularly in scenarios where the data exhibit heavy-tailed or multimodal distributions, e.g.

As crowdsourcing emerges as an efficient and cost-effective method for obtaining labels for machine learning datasets, it is important to assess the quality of crowd-provided data, so as to improve analysis performance and reduce biases in subsequent machine learning tasks. Given the lack of ground truth in most cases of crowdsourcing, we refer to data quality as annotators' consistency and credibility. Unlike the simple scenarios where Kappa coefficient and intraclass correlation coefficient usually can apply, online crowdsourcing requires dealing with more complex situations. We introduce a systematic method for evaluating data quality and detecting spamming threats via variance decomposition, and we classify spammers into three categories based on their different behavioral patterns. A spammer index is proposed to assess entire data consistency and two metrics are developed to measure crowd worker's credibility by utilizing the Markov chain and generalized random effects models. Furthermore, we showcase the practicality of our techniques and their advantages by applying them on a face verification task with both simulation and real-world data collected from two crowdsourcing platforms.

This work aims to capture the effect and benefits of depth for supervised learning via information-theoretic generalization bounds. We first derive two hierarchical bounds on the generalization error in terms of the Kullback-Leibler (KL) divergence or the 1-Wasserstein distance between the train and test distributions of the network internal representations. The KL divergence bound shrinks as the layer index increases, while the Wasserstein bound implies the existence of a layer that serves as a generalization funnel, which attains a minimal 1-Wasserstein distance. Analytic expressions for both bounds are derived under the setting of binary Gaussian classification with linear DNNs. To quantify the contraction of the relevant information measures when moving deeper into the network, we analyze the strong data processing inequality (SDPI) coefficient between consecutive layers of three regularized DNN models: Dropout, DropConnect, and Gaussian noise injection. This enables refining our generalization bounds to capture the contraction as a function of the network architecture parameters. Specializing our results to DNNs with a finite parameter space and the Gibbs algorithm reveals that deeper yet narrower network architectures generalize better in those examples, although how broadly this statement applies remains a question.

A common way of learning to perform a task is to observe how it is carried out by experts. However, it is well known that for most tasks there is no unique way to perform them. This is especially noticeable the more complex the task is because factors such as the skill or the know-how of the expert may well affect the way she solves the task. In addition, learning from experts also suffers of having a small set of training examples generally coming from several experts (since experts are usually a limited and expensive resource), being all of them positive examples (i.e. examples that represent successful executions of the task). Traditional machine learning techniques are not useful in such scenarios, as they require extensive training data. Starting from very few executions of the task presented as activity sequences, we introduce a novel inductive approach for learning multiple models, with each one representing an alternative strategy of performing a task. By an iterative process based on generalisation and specialisation, we learn the underlying patterns that capture the different styles of performing a task exhibited by the examples. We illustrate our approach on two common activity recognition tasks: a surgical skills training task and a cooking domain. We evaluate the inferred models with respect to two metrics that measure how well the models represent the examples and capture the different forms of executing a task showed by the examples. We compare our results with the traditional process mining approach and show that a small set of meaningful examples is enough to obtain patterns that capture the different strategies that are followed to solve the tasks.

The Rational Speech Act (RSA) model provides a flexible framework to model pragmatic reasoning in computational terms. However, state-of-the-art RSA models are still fairly distant from modern machine learning techniques and present a number of limitations related to their interpretability and scalability. Here, we introduce a new RSA framework for metaphor understanding that addresses these limitations by providing an explicit formula - based on the mutually shared information between the speaker and the listener - for the estimation of the communicative goal and by learning the rationality parameter using gradient-based methods. The model was tested against 24 metaphors, not limited to the conventional $\textit{John-is-a-shark}$ type. Results suggest an overall strong positive correlation between the distributions generated by the model and the interpretations obtained from the human behavioral data, which increased when the intended meaning capitalized on properties that were inherent to the vehicle concept. Overall, findings suggest that metaphor processing is well captured by a typicality-based Bayesian model, even when more scalable and interpretable, opening up possible applications to other pragmatic phenomena and novel uses for increasing Large Language Models interpretability. Yet, results highlight that the more creative nuances of metaphorical meaning, not strictly encoded in the lexical concepts, are a challenging aspect for machines.

With the widespread adoption of Large Language Models (LLMs), the prevalence of iterative interactions among these models is anticipated to increase. Notably, recent advancements in multi-round self-improving methods allow LLMs to generate new examples for training subsequent models. At the same time, multi-agent LLM systems, involving automated interactions among agents, are also increasing in prominence. Thus, in both short and long terms, LLMs may actively engage in an evolutionary process. We draw parallels between the behavior of LLMs and the evolution of human culture, as the latter has been extensively studied by cognitive scientists for decades. Our approach involves leveraging Iterated Learning (IL), a Bayesian framework that elucidates how subtle biases are magnified during human cultural evolution, to explain some behaviors of LLMs. This paper outlines key characteristics of agents' behavior in the Bayesian-IL framework, including predictions that are supported by experimental verification with various LLMs. This theoretical framework could help to more effectively predict and guide the evolution of LLMs in desired directions.